Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #120 |
| commutator | C23 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #386 |
| prime radical | C13×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
342 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,22,19,16,20,24)(17,23,26,29,25,21,27,28);(14,16,18,24,19,15,20,22)(17,23,27,28,25,21,26,29);(14,17)(15,21)(16,23)(18,27)(19,25)(20,26)(22,29)(24,28)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 623);